Numerical Analysis

Next, we provide a numerical analysis of three large-scale networks from the Stanford Net- work Analysis Project (SNAP)[142], see Fig. 2.14. Table 2.2 shows the description of the networks used. In the three cases, we select the largest strongly connected component of the graph and use it as a representative of the network structure and the mixing properties of the graph. Furthermore, we assume that the agents use equal weights for all their (in)neighbors. Table 2.2: Datasets of large-scale networks. Description, the number of nodes, the number of edges, simulated mixing time and an upper bound on the mixing time of the three datasets used in the numerical analysis. The upper bound on the mixing time is computed from the second largest eigenvalue bound in Eq. (1.4)

Graph Nodes Edges Type Upper Bound on Description

Mixing Time

wiki-Vote[143] 1300 103663 Directed 145 Wikipedia who-votes-

on-whom network

ca-GrQc [144] 4158 13428 Undirected 12308 Collaboration network of arXiv General Rel- ativity

ego-Facebook[145] 3927 88234 Undirected 53546 Social circles from Facebook

Random graph generating models, such at the Erd˝os-R´enyi graphs, the Newman-Watts graph, and the geometric random graphs, have been proposed to model the dynamics and the properties of real large-scale complex networks, for example, rapid mixing or linear

(a)

(b) (c)

Figure 2.14: Large-scale complex networks from the Stanford Network Analysis Project (SNAP). (a) The ego-Facebook, nodes are anonymized users from Facebook and edges indicate friendship status between them. (b) The wiki-Vote graph, each node represents a Wikipedia administrator and an directed edge represents a vote used for promoting a user to admin status. (c) The ca-GrQc graph is a collaboration network from arXiv authors with papers submitted to the General Relativity and Quantum Cosmology category, edges indicate co-authorship of a manuscript. The gray scale in the node colors shows the

relative social power according to the left-eigenvector corresponding to the eigenvalue 1. convergence of the beliefs. The existing approaches for the computation of such properties in real-world social networks are mainly simulation-based or require extensive computations for the approximation of the spectral properties of the graphs [146, 147]. Our analysis based on the graph-theoretical properties of the networks provides a structural explanation of the exhibited behavior. Specifically, we can explain fast mixing or equivalently linear convergence of beliefs from the existence of highly influential cliques that drive the dynamics of the complete belief system. In particular, suppose we want to study whether a specific

graph has a rapid mixing and, for example, there is a subset of ¯V of M nodes that affect the 20% of the final opinion. Then, it is enough to check if there is a finite number K such that after K time steps, a random walk in the graph has a probability of 1/5 to be in ¯V, resulting in the probability of being at any of these particularly influential nodes of 1

5M. The next theorem describes how the existence of a clique of a well-connected subset of nodes can guarantee fast mixing of a random walk on a graph.

Theorem 11. Consider a random walk on a connected undirected and static graph _G = (V, E) with _|V_| = n nodes, and assume there is a subset V¯ _⊂ V with M nodes such that after K steps, the probability of being in V¯ is at least 1₅ and the probability of being on a specific node in V¯ is at least ₅1_M. Then the mixing time of the corresponding Markov chain is of the order O(M Klog(1/)).

Proof. The proof follows immediately since any two random walks will intersect with prob- ability 1

M every K steps.

Figure 2.15 shows the cumulative influence of the nodes in each of the graphs, that is, the weight an ordered subset of the nodes has on the final value of the beliefs. In this case, since we are considering a single strongly connected component, the weights are determined by the left-eigenvalue of the weight matrix corresponding to the eigenvector 1. Table 2.3 shows the values for K and M of the graphs studied in this section.

Figure 2.16 shows the convergence time of a belief system when the network of agents is the three large-scale complex networks described in Table 2.2. Results show that the predicted maximum type behavior holds; that is, the convergence time of the belief system is upper bounded by the maximum mixing time of a random walk on the graph of agents and the graph of logic constraints. The convergence time remains constant and of the order of the convergence time of the network of agents, until the mixing time of the network formed by the logic constraints is larger. Then, the total convergence time increases based on the specific topology of the graph of logic constraints.

Table 2.3: Size of the highly influential cliques and the number of iterations required for them to drive the fast mixing of a random walk on the three examples of large scale graphs.

Graph K % of Nodes M

wiki-Vote [143] 121 10% 25

ca-GrQc[144] 365 11% 1700

0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 (20%,0.54%) Percentage of Nodes Cumm ulativ e So cial P ow er arXiv Wiki Facebook

Figure 2.15: Cumulative social power of the agents. Each of the nodes in the graphs considered has some weight in the final value achieved by the belief system. In all three cases, the 20% most important nodes account for 50% of the final value.

Figure 2.17 shows the exponential convergence rate of the belief system. It shows a linear convergence rate of the total variation distance between the beliefs and its limiting value as the number of iteration increases.