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Graph Theory
This is FIRST draft. Very likely to contain errors.
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lthough graph theory is one of the younger branches of mathematics, it is fundamental to a number of applied fields, including operations research, computer science, and social network analysis. In this chapter we discuss the basic concepts of graph theory from the point of view of social network analysis.
Graphs
The fundamental concept of graph theory is the graph, which (despite the name) is best thought of as a mathematical object rather than a diagram, even though graphs have a very natural graphical representation. A graph usually denoted G(V,E) or G = (V,E) consists of set of vertices V together with a set of edges E. Vertices are also known as nodes, points and (in social networks) as actors, agents or players. Edges are also known as lines and (in social networks) as ties or links. An edge e = (u,v) is defined by the unordered pair of vertices that serve as its end points. Two vertices u and v are adjacent if there exists an edge (u,v) that connects them. An edge e = (u,u) that links a vertex to itself is known as a self-loop or reflexive tie. The number of vertices in a graph is usually denoted n while the number of edges is usually denoted m.
As an example, the graph depicted in Figure 1 has vertex set V={a,b,c,d,e.f} and edge set E = {(a,b),(b,c),(c,d),(c,e),(d,e),(e,f)}.
Figure 1.
When looking at visualizations of graphs such as Figure 1, it is important to realize that the only information contained in the diagram is adjacency; the position of nodes in the plane (and therefore the length of lines) is arbitrary unless otherwise specified. Hence it is usually dangerous to draw conclusions based on the spatial position of the nodes. For example, it is tempting to conclude that nodes in the middle of a diagram are more important than nodes on the peripheries, but this will often if not usually be a mistake.
When used to represent social networks, we typically use each line to represent instances of the same social relation, so that if (a,b) indicates a friendship between the person located at node a and the person located at node b, then (d,e) indicates a friendship between d and e. Thus, each distinct social relation that is empirically measured on the same group of people is represented by separate graphs, which are likely to have different structures (after all, who talks to whom is not the same as who dislikes whom).
Every graph has associated with it an adjacency matrix, which is a binary n(n matrix A in which aij = 1 and aji = 1 if vertex vi is adjacent to vertex vj, and aij = 0 and aji = 0 otherwise. The natural graphical representation of an adjacency matrix is a table, such as shown in Figure 2.
abcdefa010000b101000c010110d001010e001101f000010
Figure 2. Adjacency matrix for graph in Figure 1.
Examining either Figure 1 or Figure 2, we can see that not every vertex is adjacent to every other. A graph in which all vertices are adjacent to all others is said to be complete. The extent to which a graph is complete is indicated by its density, which is defined as the number of edges divided by the number possible. If self-loops are excluded, then the number possible is n(n-1)/2. If self-loops are allowed, then the number possible is n(n+1)/2. Hence the density of the graph in Figure 1 is 6/15 = 0.40.
A clique is a maximal complete subgraph. A subgraph of a graph G is a graph whose points and lines are contained in G. A complete subgraph of G is a section of G that is complete (i.e., has density = 1). A maximal complete subgraph is a subgraph of G that is complete and is maximal in the sense that no other node of G could be added to the subgraph without losing the completeness property. In Figure 1, the nodes {c,d,e} together with the lines connecting them form a clique. Cliques have been seen as a way to represent what social scientists have called primary groups.
While not every vertex in the graph in Figure 1 is adjacent, one can construct a sequence of adjacent vertices from any vertex to any other. Graphs with this property are called connected. Similarly, any pair of vertices in which one vertex can reach the other via a sequence of adjacent vertices is called reachable. If we determine reachability for every pair of vertices, we can construct a reachability matrix R such as depicted in Figure 3. The matrix R can be thought of as the result of applying transitive closure to the adjacency matrix A.
Figure 3.
A component of a graph is defined as a maximal subgraph in which a path exists from every node to every other (i.e., they are mutually reachable). The size of a component is defined as the number of nodes it contains. A connected graph has only one component.
A sequence of adjacent vertices v0,v1,,vn is known as a walk. In Figure 3, the sequence a,b,c,b,a,c is a walk. A walk can also be seen as a sequence of incident edges, where two edges are said to be incident if they share exactly one vertex. A walk in which no vertex occurs more than once is known as a path. In Figure 3, the sequence a,b,c,d,e,f is a path. A walk in which no edge occurs more than once is known as a trail. In Figure 3, the sequence a,b,c,e,d,c,g is a trail but not a path. Every path is a trail, and every trail is a walk. A walk is closed if vo = vn. A cycle can be defined as a closed path in which n >= 3. The sequence c,e,d in Figure 3 is a cycle. A tree is a connected graph that contains no cycles. In a tree, every pair of points is connected by a unique path. That is, there is only one way to get from A to B.
The length of a walk (and therefore a path or trail) is defined as the number of edges it contains. For example, in Figure 3, the path a,b,c,d,e has length 4. A walk between two vertices whose length is as short as any other walk connecting the same pair of vertices is called a geodesic. Of course, all geodesics are paths. Geodesics are not necessarily unique. From vertex a to vertex f in Figure 1, there are two geodesics: a,b,c,d,e,f and a,b,c,g,e,f.
The graph-theoretic distance (usually shortened to just distance) between two vertices is defined as the length of a geodesic that connects them. If we compute the distance between every pair of vertices, we can construct a distance matrix D such as depicted in Figure 4. The maximum distance in a graph defines the graphs diameter. As shown in Figure 4, the diameter of the graph in Figure 1 is 4. If the graph is not connected, then there exist pairs of vertices that are not mutually reachable so that the distance between them is not defined and the diameter of such a graph is also not defined.
abcdefga0123343b1012232c2101121d3210122e3211011f4322102g3212120
Figure 4. Distance matrix for graph in Figure 3.
The powers of a graphs adjacency matrix, Ap, give the number of walks of length p between all pairs of nodes. For example, A2, obtained by multiplying the matrix by itself, has entries EMBED Equation.3 that give the number of walks of length 2 that join node vi to node vj. Hence, the geodesic distance matrix D has entries dij = p, where p is the smallest p such that EMBED Equation.3 > 0. (However, there exist much faster algorithms for computing the distance matrix.)
The eccentricity e(v) of a point v in a connected graph G(V,E) is max d(u,v), for all u ( V. In other words, a points eccentricity is equal to the distance from itself to the point farthest away. The eccentricity of node b in Figure 3 is 3. The minimum eccentricity of all points in a graph is called the radius r(G) of the graph, while the maximum eccentricity is the diameter of the graph. In Figure 3, the radius is 2 and the diameter is 4. A vertex that is least distant from all other vertices (in the sense that its eccentricity equals the radius of the graph) is a member of the center of the graph and is called a central point. Every tree has a center consisting of either one point or two adjacent points.
The number of vertices adjacent to a given vertex is called the degree of the vertex and is denoted d(v). It can be obtained from the adjacency matrix of a graph by simply computing each row sum. For example, the degree of vertex c in Figure 3 is 4. The average degree, EMBED Equation.3 , of all vertices depicted in Figure 3 is 2.29. There is a direct relationship between the average degree, EMBED Equation.3 , of all vertices in a graph and the graphs density:
EMBED Equation.3
The minimum degree of a graph G is denoted ((G). A vertex with degree 0 is known as an isolate (and constitutes a component of size 1), while a vertex with degree 1 is a pendant. Holding average degree constant, there is a tendency for graphs that contain some nodes of high degree (i.e., high variance in degree) to have shorter distances than graphs with lower variance, with the high degree nodes serving as shortcuts across the network.
A node whose removal from a graph disconnects the graph (or, more generally, increases the number of components in the graph) is called a cutpoint or an articulation point. The graph in Figure 3 has three cutpoints, namely b, c, and e. A connected, non-trivial graph is called non-separable if it has no cutpoints. A block or bi-component is a maximal nonseparable subgraph. Blocks partition the edges in a graph into mutually exclusive edges. They also share no nodes except cutpoints. Thus, cutpoints decompose graphs into (nearly) non-overlapping sections. In blocks of more than two points, every pair of points lies along a common cycle, which means that there is always a minimum of two ways to get from any point to any other. In Figure 3, we find the following blocks: {a,b}, {b,c}, {c,d,e,g}, {e,f}.
The notion of a cutpoint can be generalized to a cutset, which is a set of points whose joint removal increases the number of components in the graph. Of particular interest is a minimum weight cutset, which is a cutset that is as small as possible (i.e., no other cutset has fewer members). There can be more than one distinct minimum weight cutset in a graph. The size of a graph s minimum weight cutset defines the vertex connectivity (G) of a graph, which is the minimum number of nodes that must be removed to increase the number of components in the graph (or render it trivial). The point connectivity of a disconnected graph is 0. The point connectivity of a graph containing a cutpoint is no higher than 1. The point connectivity of a non-separable graph is at least 2. We can analogously define the vertex connectivity (u,v) of a pair of points u,v as the number of nodes that must be removed to disconnect that pair. The connectivity of the graph (g) is just the minimum (u,v) for all u,v in V.
A famous theorem by Menger published in 1929 relates the vertex connectivity of a pair of nodes to the maximum number of node-independent paths connecting those nodes. A set of paths from a source node s to a target node t is node-independent if none of the paths share any vertices aside from s and t. Mengers theorem states that for any source s and target t, the maximum number of node-independent paths between s and t is equal to the vertex connectivity of that pair i.e., the number of nodes that must be removed to disconnect them. Hence, there might be many different paths from s to t, but if they all share a certain node (i.e., are not independent), then s and t can easily be disconnected by eliminating just that node.
Thus, we can think of the point connectivity of a graph as an indicator of the invulnerability of the graph to threats of disconnection by removal of nodes. If (G) is high, or if the average (u,v) is high for all pairs of nodes, then we know that it is fairly difficult to disconnect the nodes in the graph by removing intermediaries.
The vertex-based notions of cutpoint, cutset, vertex connectivity and node-independent path set have analogous counterparts for edges. A bridge is defined as an edge whose removal would increase the number of components in the graph. Edge connectivity is denoted ((G) and the edge connectivity of a pair of nodes is denoted ((u,v). A disconnected graph has ((G)=0, while a graph with a bridge has ((G)=1. Point connectivity and line connectivity are related to each other and to the minimum degree in a graph by Whitneys inequality:
EMBED Equation.3
Directed Graphs
As noted at the outset, the edges contained in graphs are unordered pairs of nodes (i.e., (u,v) is the same thing as (v,u)). As such, graphs are useful for encoding directionless relationships such as the social relation sibling of or the physical relation is near. However, many relations that we would like to model are not directionless. For example, is the boss of is usually anti-symmetric in the sense that if u is the boss of v, it is unlikely that v is the boss of u. Other relations, such as gives advice to are simply non-symmetric in the sense that if u gives advice to v, v may or may not give advice to u.
To model non-symmetric relations we use directed graphs, also known as digraphs. A digraph D(V,E) consists of a set of nodes V and a set of ordered pairs of nodes E called arcs or directed lines. The arc (u,v) points from u to v.
Digraphs are usually represented visually like graphs, except that arrowheads are placed on lines to indicate direction (see Figure 5). When both arcs (u,v) and (v,u) are present in a digraph, they may be represented by a double-headed arrow (as in Figure 5a), or two separate arrows (as shown in Figure 5b).
EMBED MetaSoftwareMetaDesign4.0 Figure 5a EMBED MetaSoftwareMetaDesign4.0 Figure 5b
In a digraph, a walk is a sequence of nodes vo,v1,vn in which each pair of nodes vi, vi+1 is linked by an arc (vi,vi+1). In other words, it is a traversal of the graph in which the flow of movement follows the direction of the arcs, like a car moving from place to place via one-way streets. A path in a digraph is a walk in which all points are distinct. A semiwalk is a sequence of nodes vo,v1,vn in which each pair of nodes vi, vi+1 is linked by either the arc (vi,vi+1) or the arc (vi+1,vi). In other words, in a semiwalk, the traversal need not respect the direction of arcs, like a car that freely goes the wrong way on one-way streets. By analogy, we can also define a semipath, semitrail, and semicycle.
Another way to think of semiwalks is as walks on the underlying graph, where the underlying graph is the graph G(V,E) that is formed from the digraph D(V,E) such that (u,v) ( E if and only if (u,v) ( E or (v,u) ( E. Thus, the underlying graph of a digraph is basically the graph formed by ignoring directionality.
A digraph is strongly connected if there exists a path (not a semipath) from every point to every other. Note that the path from u to v need not involve the same intermediaries as the path from v to u. A digraph is unilaterally connected if for every pair of points there is a path from one to the other (but not necessarily the other way around). A digraph is weakly connected if every pair of points is mutually reachable via a semipath (i.e., if the underlying graph is connected).
A strong component of a digraph is a maximal strongly connected subgraph. In other words, it is a subgraph that is strongly connected and which is as large as possible (there is no node outside the subgraph that is strongly connected to all the nodes in the subgraph). A weak component is a maximal weakly connected subgraph.
The number of arcs originating from a node v (i.e., outgoing arcs) is called the outdegree of v, denoted od(v). The number of arcs pointing to a node v (i.e., incoming arcs) is called the indegree of v, denoted id(v). In a graph representing friendship feelings among a set of persons, outdegree can be seen as indicating gregariousness, while indegree corresponds to popularity. The average outdegree of a digraph is necessarily equal to the average indegree.
The adjacency matrix A of a digraph is an n n matrix in which aij = 1 if (vi,vj) ( E and aij = 0 otherwise. Unlike the adjacency matrix of an undirected graph, the adjacency matrix of a directed graph is not constrained to be symmetric, so that the top right half need not equal the bottom left half (i.e., aij <> aji). If a digraph is acyclic, then it is possible to order the points of D so that the adjacency matrix upper triangular (i.e., all positive entries are above the main diagonal).
Social Network Extensions to Graph Theory
In this section we consider contributions to graph theory from the study of social networks. There are two main groups of contributions: cohesive subsets and roles/positions. Note that the definitions of cohesive subsets assume graphs, while those of roles/position assume digraphs.
Cohesive Subsets
It was mentioned earlier that the notion of a clique can be seen as formalizing the notion of a primary group. A problem with this, however, is that it is too strict to be practical: real groups will contain several pairs of people who dont have a close relationship. A relaxation and generalization of the clique concept is the n-clique. An n-clique S of a graph is a maximal set of nodes in which for all u,v ( S, the graph-theoretic distance d(u,v) <= n. In other words, an n-clique is a set of nodes in which every node can reach every other in n or fewer steps, and the set is maximal in the sense that no other node in the graph is distance n or less from every other node in the subgraph. A 1-clique is the same as an ordinary clique. The set {a,b,c,d,e} in Figure 6 is an example of a 2-clique.
EMBED MetaSoftwareMetaDesign4.0 Figure 6.
Note that the path of length n or less linking a member of the n-clique to another member may pass through an intermediary who is not in the group. In the 2-clique in Figure 6, nodes c and e are distance 2 only because of d, which is not a member of the 2-clique. In this sense, n-cliques are not as cohesive as they might otherwise appear. The notion of an n-clan avoids that. An n-clan is an n-clique in which the diameter of the subgraph G induced by S is less than or equal to n. The subgraph G of a graph G induced by the set of nodes S is defined as the maximal subgraph of G that has point set S. In other words, it is the subgraph of G obtained by taking all nodes in S and all ties among them. Therefore, an n-clan S is an n-clique in which all pairs have distance less than or equal to n even when we restrict all paths to involve only members of S. In Figure 6, the set {b,c,d,e,f} is a 2-clan, but {a,b,c,d,e} is not because b and c have distance greater than 2 in the induced subgraph. Note that {a,b,f,e} is also fails the 2-clan criterion because n-clans are defined to be n-cliques and {a,b,f,e} is not a 2-clique (it fails the maximality criterion since {a,b,c,d,e}). An n-club corrects this problem by eliminating the n-clique criterion from the definition. An n-club is a subset S of nodes such that in the subgraph induced by S, the diameter is n or less. Every n-clan is both an n-club and an n-clique. The set {a,b,c,f} is a 2-club.
Whereas n-cliques, n-clans and n-clubs all generalize the notion of clique via relaxing distance, the k-plex generalizes the clique by relaxing density. A k-plex is a subset S of nodes such that every member of the set is connected to n-k others, where n is the size of S. Although not part of the official definition, it is conventional to additionally impose a maximality condition, so that proper subsets of k-plexes are ignored. There are some guarantees on the cohesiveness of k-plexes. For example, k-plexes in which k < (n+2)/2 have no distances greater than 2 and cannot contain bridges (making them resistant to attack by deleting an edge). In Figure 6, the set {a,b,c,f} fails to be a 2-plex because each member must have at least 4-2=2 ties to other members of the set, yet c has only one tie within the group. In the graph in Figure 7, the set {a,b,d,e} is a 2-plex.
EMBED MetaSoftwareMetaDesign4.0 Figure 7.
More cohesive than k-plexes are LS sets. Let H be a set of nodes in graph G(V,E) and let K be a proper subset of H. Let (K) denote the number of edges linking members of K to V-K (the set of nodes not in K). Then H is an LS set of G if for every proper subset K of H, (K) > (H). The basic idea is that individuals in H have more ties with other members than they do to outsiders. Another way to define LS sets that makes this more evident is as follows. Let (X,Y) denote the number of edges from members of set X to members of set Y. Then H is an LS set if (K,H-K) > (K,V-H). In Figure 7, the set {a,b,d,e} is not an LS set since ({b,d,e},{a}) is not greater than ({b,d,e},{c}). In contrast, the set {a,b,d,e} in Figure 8 does qualify as an LS set.
EMBED MetaSoftwareMetaDesign4.0 Figure 8.
A key property of LS sets is high edge connectivity. Specifically, every node in an LS set has higher edge connectivity (() with other members of the LS set than with any non-member. Taking this as the sole criterion for defining a cohesive subset, a lambda set is defined as a maximal subset of nodes S such that for all a,b,c ( S and d ( V-S, ((a,b) > ((c,d). To the extent that ( is high, members of the same lambda set are difficult to disconnect from one another because ( defines the number edges that must be removed from the graph in order to disconnect the nodes within the lambda set.
A k-core is a maximal subgraph H in which (H) >= k. Hence, every member of a 2-core is connected to at least 2 other members, and no node outside the 2-core is connected to 2 or more members of the core (otherwise it would not be maximal). Every k-core contains at least k+1 vertices, and vertices in different k-cores cannot be adjacent. A 1-core is simply a component. K-cores can be described as loosely cohesive regions which will contain more cohesive subsets. For example, every k-plex is contained in a k-core.
Roles/Positions
Given a digraph D(V,E), the in-neighborhood of a node v, denoted Ni(v) is the set of vertices that send arcs to v. That is, Ni(v) = {u: (u,v) ( E}. The out-neighborhood of a node v, denoted No(v) is the set of vertices that receive arcs from v. That is, No(v) = {u: (v,u) ( E}.
A coloration C is an assignments of colors to the vertices V of a digraph. The color of a vertex v is denoted C(v) and the set of distinct colors assigned to nodes in a set S is denoted C(S) and termed the spectrum of S. In Figure 9, a coloration of nodes is depicted by labeling the nodes with letters such as r for red, and y for yellow. Nodes colored the same are said to be equivalent.
EMBED MetaSoftwareMetaDesign4.0 Figure 9.
A coloration is a strong structural coloration if nodes are assigned the same color if and only if they have identical in and out neighborhoods. That is, for all u,v (V, C(u) = C(v) if and only if Ni(u) = Ni(v) and No(u) = No(v). The coloration in Figure 9 is a strong structural coloration. We can check this by taking pairs of nodes and verifying that if they are colored the same (i.e., are strongly structurally equivalent) they have identical neighborhoods, and if they are not colored the same, they have different neighborhoods. For example, b and d are colored the same, and both of their neighborhoods consist of {a,c,e}.
Note that in strong structural colorations, any two nodes that are colored the same are structurally identical: if we remove the identifying labels from the identically colored nodes, then spin the graph around in space before placing it back down on the page, we would not be able to figure out which of the same-colored nodes was which. Consequently, any property of the nodes that stems from their structural position (such as expected time until arrival of something flowing through the network) should be the same for nodes that are equivalent.
A coloration C is regular if C(u) = C(v) implies that C(Ni(u)) = C(Ni(v)) and C(No(u)) = C(No(v)) for all u, v ( V. In other words, in regular colorations, every pair of nodes that has the same color must receive arcs from nodes comprising the same set of colors and must send arcs to nodes comprising the same set of colors. Every structural coloration is a regular coloration.
EMBED MetaSoftwareMetaDesign4.0 Figure 10. Regular coloration.
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If a graph represents a social network, we can think of the colors as defining emergent classes or types of people such that if one member of a certain class (blue) has outgoing ties to members of exactly two other classes (yellow and green), then all other members of that (blue) class have outgoing ties to members of those same two classes (yellow and green). Thus, regular colorations classify members of a social network according to their pattern of relations of others, and two people are placed in the same class if they interact in the same ways with the same kinds of others (but not necessarily with same individuals).
Just as the various generalizations of cliques are attempts to capture mathematically the notion of a social group, regular colorations are an attempt to capture the notion of a social role system, in which people playing a certain role have characteristic relations with others who are playing complementary roles (such as doctors, nurses and patients).
Cohesive subsets are traditionally defined in terms of subgraphs rather than subsets of nodes. However, since most people think about them in terms of node sets, and because using subgraphs complicates notation, we used subsets here.
1st Draft, written very quickly. May contain errors. Be aware.
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