Fig. 1
From: Managing large distributed dynamic graphs for smart city network applications
A schematic global smart city graph breaching caused by a local breach. The attribute which is handled in a graph G is \(\overline {deg}(G(V))\)– the vertices average degree (number of edges incident to a vertex). a At some starting point \(t=0 \overline {deg}(D_{1}(V))|_{t=0}=2.5, \overline {deg}(D_{2}(V))|_{t=0}=2.4\) and \(\overline {deg}(D_{3}(V))|_{t=0}=2.67\), giving \(\overline {deg}(G(V))|_{t=0}=2.52\), which is greater than the global graph threshold T preliminary set to be ≥ 1.6. b A change occurs locally at t= 1min for which \(\overline {deg}(D_{1}(V))|_{t=1}=1\), meaning several edges in D1 (B, C) were removed and the local threshold T1 was breached, however not the global T, since still \(\overline {deg}(G(V))|_{t=1}=1.97>1.6\). c A breach of T3 occurred at t=2, in which \(\overline {deg}(D_{3}(V))|_{t=2}=1.33\), i.e., several edges (J, K, L) in D3 were removed. In this case the global T breaches since \(\overline {deg}(G(V))|_{t=2}=1.57<1.6\)